# Intuition of Homology

$\cdots \longrightarrow C_{n+1} \overset{\partial_{n+1}} {\longrightarrow} C_n \overset{\partial_n}{\longrightarrow} C_{n-1} \overset{\partial_{n-1}}{\longrightarrow}C_{n-2}\longrightarrow\cdots$

Homology: $H_n = \text{Ker } \partial_n/ \text{Im }\partial_{n+1}$

Intuition of Homology: $C_n$ is some $n-$ dimensional object, and $\partial_n$ gets its boundary. $\partial_{n-1}\circ \partial_{n}=0$ suggests a boundary’s boundary is $\emptyset$. $\text{Ker } \partial_{n-1}$ is the $(n-1)-$ dimensional things that no more has a boundary. $\text{Im }\partial_n$ is the $(n-1)-$ dimensional things that is a boundary (of some $n-$ dimensional thing). Now from this, it is still hard to visualize what $H_{n-1} = \text{Ker } \partial_{n-1}/ \text{Im }\partial_n$ is. This suggests that homology does not have a general intuition.

Instead, let’s look at the simplest example in simplicial homology: a $\Delta_2$ labeled  by its three vertices, $a,b,c$. We have $\text{Im }\partial_2 = n(ab+bc+ca)$, $\text{Ker }\partial_1 = n(ab+bc+ca)$, thus $H_1 = 0$. On the other hand, if we remove the face $abc$ leaving only the skeleton, $\text{Im }\partial_2=0$, thus $H_1 = \{n\}/\emptyset = \mathbb{Z}$. We see that $\text{Ker } \partial_{n-1}$ is like “the boundary of something”, and $latex \text{Im }\partial_n$ is like  “is this ‘something’ filled or not”? if it is filled, then we have no hole. if it is not filled, then we have a hole. This hole is captured by $H_{n-1}$. However, this intuition is only true for the $\mathbb{Z}$ part of the homology (one $\mathbb{Z}$ corresponds to one hole) and has not accounted for (e.g.) $\mathbb{Z}/2\mathbb{Z}$ being a homology group.

In fact, finite part $\mathbb{Z}/2\mathbb{Z}$ shows the degree of non-triviality a hole is. If it is just an ordinary hole, we can wrap it around as many times as we want so we get a $\mathbb{Z}$. However, if a hole is nontrivial meaning some wrapping actually deforms to identity (no wrapping at all), then we get some finite part like $\mathbb{Z}/2\mathbb{Z}$. An example would be $\mathbb{R}P_n$.